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The surface of the torus is spatially flat, like the piece of paper, but finite. However, with expansion, it is possible that even if the universe just has a very large volume now, it will reach infinite volume in the infinite future.

You have to define what you mean by infinite. Infinite mass? Infinite volume? And also how you would measure these things. Infinite radius, for instance. If we accept relativity, you can never reach the end of the universe, even if the current radius is finite. So, is it meaningfully finite?

In a way you could argue that the universe is infinite because its rate of expansion is faster than the rate at which one can travel within it. Thus you can travel infinitely far in any direction without reaching the edge.